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A383922 a(n) = A002104(n) + A002104(n+1) - 1.

%I A383922 #24 May 20 2025 21:44:32
%S A383922 0,3,10,31,112,503,2786,18443,141744,1237755,12088266,130457479,
%T A383922 1541023936,19769882767,273671845058,4065274481939,64493941507232,
%U A383922 1088226653465139,19458541429154250,367527663494842671,7311506648705326672,152804399672163086695,3347034732868985727202,76675452816691696778843
%N A383922 a(n) = A002104(n) + A002104(n+1) - 1.
%C A383922 Let m_0(x) = -x for x < 0 and (1/2)*m_0(x - m_0(x-1)) for x >= 0, then:
%C A383922    m_0(1 - 2^(-n)) = 2^(-(n+1)) for n >= -1;
%C A383922    m_0(2 - 2^(-n)) = 2^(-(2*n+3)) for n >= -2;
%C A383922    m_0(3 - 2^(-n)) = 2^(-a(n+2)) for n >= -2 (see my link for a proof).
%C A383922 Let F_0 = {x + m_0(x) : x in R}, then the intersection of F_0 and (-oo,n) is well-ordered with order type omega^^n. As a result, F_0 is well-ordered with order type epsilon_0 = omega^^omega. F_0 is a proper subset of F, the set of fusible numbers.
%C A383922 It had been believed that x + m_0(x) was the least fusible greater than x. _Junyan Xu_ points out that this is false. Indeed, let x_n = 17/8 - 1/2^(n+1), c_n = 5/4 - 1/2^(n+1), and d_n = 2 - 1/2^(n-2) + 1/2^(2*n-1) for n >= 0, then
%C A383922     c_n = (1/2 + (1-1/2^n) + 1)/2,
%C A383922     d_n = ((1-1/2^(2*n-2)) + d_{n-1} + 1)/2, n >= 1
%C A383922   are both fusible numbers, hence so is (c_n + d_{n+3} + 1)/2 = 17/8 - 1/2^(n+1) + 1/2^(2*n+6); in other words, the least fusible number greater than x_n is at most x_n + 1/2^(2*n+6). But we have m_0(x_n) = 1/2^(n+8) > 1/2^(2*n+6) for n >= 3.
%C A383922 _Junyan Xu_ gives a conjecture on the recursive formula of m(x), where x + m(x) is the least fusible greater than x. (A188545(n) is -log_2 m(n)). We have m(x) = m_0(x) for x < 33/16 (i.e., F_0 and F coincide on the interval (-oo,33/16]), but they differ for x >= 33/16, even if the intersection of F and (-oo,n) still has order type omega^^n if the conjecture is true. In fact, we have -log_2 m_0(3) = 1541023937, while -log_2 m(3) > 2^^^^^^^^^16 in Knuth's up-arrow notation.
%H A383922 Jianing Song, <a href="/A383922/a383922.pdf">Notes on A383922</a>
%H A383922 Junyan Xu, <a href="https://arxiv.org/abs/1202.5614">Survey on Fusible Numbers</a>, arXiv:1202.5614 [math.CO], 2012.
%F A383922 E.g.f.: exp(x) * (-2*log(1-x) + x/(1-x)).
%F A383922 E.g.f. satisfies (1-x)^2 * (A'(x) - A(x)) = (-2*x+3)*exp(x).
%F A383922 Recurrence: n*a(n) = (n^2+n-1)*a(n-1) - (n^2-1)*a(n-2) + 2*n + 1, a(0) = 0, a(1) = 3.
%F A383922 Recurrence: a(n) = n*a(n-1) - a(n-2) - (n-2)*a(n-3) + 4, a(0) = 0, a(1) = 3, a(2) = 10.
%F A383922 a(n) ~ exp(1)*n!.
%o A383922 (PARI) a(n) = A002104(n) + A002104(n+1) - 1 \\ See A002104
%Y A383922 Cf. A002104, A188545.
%K A383922 nonn
%O A383922 0,2
%A A383922 _Jianing Song_, May 15 2025